Polytope of Type {4,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,14}*672b
if this polytope has a name.
Group : SmallGroup(672,1260)
Rank : 4
Schlafli Type : {4,6,14}
Number of vertices, edges, etc : 4, 12, 42, 14
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,14,2} of size 1344
Vertex Figure Of :
   {2,4,6,14} of size 1344
Quotients (Maximal Quotients in Boldface) :
   7-fold quotients : {4,6,2}*96c
   14-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,14}*1344b, {4,12,14}*1344c, {4,6,28}*1344b, {4,6,14}*1344
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168);;
s1 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)( 29, 57)
( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)( 37, 65)
( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)( 45, 73)
( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)( 53, 81)
( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)(102,103)
(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)(118,147)
(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)(126,155)
(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)(134,163)
(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);;
s2 := (  1, 57)(  2, 60)(  3, 59)(  4, 58)(  5, 81)(  6, 84)(  7, 83)(  8, 82)
(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 73)( 14, 76)( 15, 75)( 16, 74)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 30, 32)( 33, 53)( 34, 56)( 35, 55)
( 36, 54)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 85,141)( 86,144)( 87,143)( 88,142)( 89,165)( 90,168)( 91,167)
( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)( 97,157)( 98,160)( 99,159)
(100,158)(101,153)(102,156)(103,155)(104,154)(105,149)(106,152)(107,151)
(108,150)(109,145)(110,148)(111,147)(112,146)(114,116)(117,137)(118,140)
(119,139)(120,138)(121,133)(122,136)(123,135)(124,134)(125,129)(126,132)
(127,131)(128,130);;
s3 := (  1, 89)(  2, 90)(  3, 91)(  4, 92)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9,109)( 10,110)( 11,111)( 12,112)( 13,105)( 14,106)( 15,107)( 16,108)
( 17,101)( 18,102)( 19,103)( 20,104)( 21, 97)( 22, 98)( 23, 99)( 24,100)
( 25, 93)( 26, 94)( 27, 95)( 28, 96)( 29,117)( 30,118)( 31,119)( 32,120)
( 33,113)( 34,114)( 35,115)( 36,116)( 37,137)( 38,138)( 39,139)( 40,140)
( 41,133)( 42,134)( 43,135)( 44,136)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,125)( 50,126)( 51,127)( 52,128)( 53,121)( 54,122)( 55,123)( 56,124)
( 57,145)( 58,146)( 59,147)( 60,148)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,165)( 66,166)( 67,167)( 68,168)( 69,161)( 70,162)( 71,163)( 72,164)
( 73,157)( 74,158)( 75,159)( 76,160)( 77,153)( 78,154)( 79,155)( 80,156)
( 81,149)( 82,150)( 83,151)( 84,152);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(168)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168);
s1 := Sym(168)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)
( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)
( 37, 65)( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)
( 45, 73)( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)
( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)
(102,103)(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)
(118,147)(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)
(126,155)(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)
(134,163)(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);
s2 := Sym(168)!(  1, 57)(  2, 60)(  3, 59)(  4, 58)(  5, 81)(  6, 84)(  7, 83)
(  8, 82)(  9, 77)( 10, 80)( 11, 79)( 12, 78)( 13, 73)( 14, 76)( 15, 75)
( 16, 74)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)
( 24, 66)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 30, 32)( 33, 53)( 34, 56)
( 35, 55)( 36, 54)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 45)( 42, 48)
( 43, 47)( 44, 46)( 85,141)( 86,144)( 87,143)( 88,142)( 89,165)( 90,168)
( 91,167)( 92,166)( 93,161)( 94,164)( 95,163)( 96,162)( 97,157)( 98,160)
( 99,159)(100,158)(101,153)(102,156)(103,155)(104,154)(105,149)(106,152)
(107,151)(108,150)(109,145)(110,148)(111,147)(112,146)(114,116)(117,137)
(118,140)(119,139)(120,138)(121,133)(122,136)(123,135)(124,134)(125,129)
(126,132)(127,131)(128,130);
s3 := Sym(168)!(  1, 89)(  2, 90)(  3, 91)(  4, 92)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9,109)( 10,110)( 11,111)( 12,112)( 13,105)( 14,106)( 15,107)
( 16,108)( 17,101)( 18,102)( 19,103)( 20,104)( 21, 97)( 22, 98)( 23, 99)
( 24,100)( 25, 93)( 26, 94)( 27, 95)( 28, 96)( 29,117)( 30,118)( 31,119)
( 32,120)( 33,113)( 34,114)( 35,115)( 36,116)( 37,137)( 38,138)( 39,139)
( 40,140)( 41,133)( 42,134)( 43,135)( 44,136)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,125)( 50,126)( 51,127)( 52,128)( 53,121)( 54,122)( 55,123)
( 56,124)( 57,145)( 58,146)( 59,147)( 60,148)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,165)( 66,166)( 67,167)( 68,168)( 69,161)( 70,162)( 71,163)
( 72,164)( 73,157)( 74,158)( 75,159)( 76,160)( 77,153)( 78,154)( 79,155)
( 80,156)( 81,149)( 82,150)( 83,151)( 84,152);
poly := sub<Sym(168)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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